Ramification of the eigencurve at classical RM points

J.Bellaïche and M.Dimitrov have shown that the $p$-adic eigencurve is smooth but not etale over the weight space at $p$-regular theta series attached to a character of a real quadratic field $F$ in which $p$ splits. We proof in this paper the existence of an isomorphism between the subring of the completed local ring of the eigencurve at these points fixed by the Atkin-Lehner involution and an universal ring representing a pseudo-deformation problem, and one gives also a precise criterion for which the ramification index is exactly $2$. We finish this paper by proving the smoothness of the nearly ordinary and ordinary Hecke algebras for Hilbert modular forms over $F$ at the cuspidal-overconvergent Eisenstein points which are the base change lift for $\mathrm{GL}(2)_{/F}$ of these theta series.

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